Long-range random walk on percolation clusters.

Abstract

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Random walks on square-lattice percolation clusters are simulated for interaction ranges spanning one to five nearest-neighbor bonds (R=1 , . . . , 5). The relative hopping probability is given by exp(-\ensuremath{\alpha}r), where r is the number of bonds traversed in one hop and \ensuremath{\alpha} is a parameter (0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}10). The fractal exponent for the random walks is universal. For R=2 (and R=1) we obtain a spectral dimension of ${d}_{s}$=1.31\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the Alexander-Orbach conjecture (1.333), and in even better agreement with the Aharony-Stauffer conjecture (1.309). Our results are based on the relation ${d}_{s}$=(91/43)f, where ${S}_{n}$\ensuremath{\sim}${N}^{f}$ describes the mean number (${S}_{N}$) of distinct sites visited in N steps for walks originating on all clusters. While the asymptotic limit of f is closely approached after 5000 nominal time steps for \ensuremath{\alpha}=0, much longer times (>50 000 steps) are required for \ensuremath{\alpha}\ensuremath{\gg}0. We also observe fractal-to-Euclidean crossovers above criticality; again, this crossover takes much longer for \ensuremath{\alpha}\ensuremath{\gg}0.